4,945 research outputs found
Optimal queue-size scaling in switched networks
We consider a switched (queuing) network in which there are constraints on
which queues may be served simultaneously; such networks have been used to
effectively model input-queued switches and wireless networks. The scheduling
policy for such a network specifies which queues to serve at any point in time,
based on the current state or past history of the system. In the main result of
this paper, we provide a new class of online scheduling policies that achieve
optimal queue-size scaling for a class of switched networks including
input-queued switches. In particular, it establishes the validity of a
conjecture (documented in Shah, Tsitsiklis and Zhong [Queueing Syst. 68 (2011)
375-384]) about optimal queue-size scaling for input-queued switches.Comment: Published in at http://dx.doi.org/10.1214/13-AAP970 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Resource allocation in stochastic processing networks : performance and scaling
Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 189-193).This thesis addresses the design and analysis of resource allocation policies in largescale stochastic systems, motivated by examples such as the Internet, cloud facilities, wireless networks, etc. A canonical framework for modeling many such systems is provided by "stochastic processing networks" (SPN) (Harrison [28, 29]). In this context, the key operational challenge is efficient and timely resource allocation. We consider two important classes of SPNs: switched networks and bandwidth-sharing networks. Switched networks are constrained queueing models that have been used successfully to describe the detailed packet-level dynamics in systems such as input-queued switches and wireless networks. Bandwidth-sharing networks have primarily been used to capture the long-term behavior of the flow-level dynamics in the Internet. In this thesis, we develop novel methods to analyze the performance of existing resource allocation policies, and we design new policies that achieve provably good performance. First, we study performance properties of so-called Maximum-Weight-[alpha] (MW-[alpha]) policies in switched networks, and of a-fair policies in bandwidth-sharing networks, both of which are well-known families of resource allocation policies, parametrized by a positive parameter [alpha] > 0. We study both their transient properties as well as their steady-state behavior. In switched networks, under a MW-a policy with a 2 1, we obtain bounds on the maximum queue size over a given time horizon, by means of a maximal inequality derived from the standard Lyapunov drift condition. As a corollary, we establish the full state space collapse property when [alpha] > 1. In the steady-state regime, for any [alpha] >/= 0, we obtain explicit exponential tail bounds on the queue sizes, by relying on a norm-like Lyapunov function, different from the standard Lyapunov function used in the literature. Methods and results are largely parallel for bandwidth-sharing networks. Under an a-fair policy with [alpha] >/= 1, we obtain bounds on the maximum number of flows in the network over a given time horizon, and hence establish the full state space collapse property when [alpha] >/= 1. In the steady-state regime, using again a norm-like Lyapunov function, we obtain explicit exponential tail bounds on the number of flows, for any a > 0. As a corollary, we establish the validity of the diffusion approximation developed by Kang et al. [32], in steady state, for the case [alpha] = 1. Second, we consider the design of resource allocation policies in switched networks. At a high level, the central performance questions of interest are: what is the optimal scaling behavior of policies in large-scale systems, and how can we achieve it? More specifically, in the context of general switched networks, we provide a new class of online policies, inspired by the classical insensitivity theory for product-form queueing networks, which admits explicit performance bounds. These policies achieve optimal queue-size scaling, in the conventional heavy-traffic regime, for a class of switched networks, thus settling a conjecture (documented in [51]) on queue-size scaling in input-queued switches. In the particular context of input-queued switches, we consider the scaling behavior of queue sizes, as a function of the port number n and the load factor [rho]. In particular, we consider the special case of uniform arrival rates, and we focus on the regime where [rho] = 1 - 1/f(n), with f(n) >/= n. We provide a new class of policies under which the long-run average total queue size scales as O(n1.5 -f(n) log f(n)). As a corollary, when f(n) = n, the long-run average total queue size scales as O(n2.5 log n). This is a substantial improvement upon prior works [44], [52], [48], [39], where the same quantity scales as O(n3 ) (ignoring logarithmic dependence on n).by Yuan Zhong.Ph.D
Switched networks with maximum weight policies: Fluid approximation and multiplicative state space collapse
We consider a queueing network in which there are constraints on which queues
may be served simultaneously; such networks may be used to model input-queued
switches and wireless networks. The scheduling policy for such a network
specifies which queues to serve at any point in time. We consider a family of
scheduling policies, related to the maximum-weight policy of Tassiulas and
Ephremides [IEEE Trans. Automat. Control 37 (1992) 1936--1948], for single-hop
and multihop networks. We specify a fluid model and show that fluid-scaled
performance processes can be approximated by fluid model solutions. We study
the behavior of fluid model solutions under critical load, and characterize
invariant states as those states which solve a certain network-wide
optimization problem. We use fluid model results to prove multiplicative state
space collapse. A notable feature of our results is that they do not assume
complete resource pooling.Comment: Published in at http://dx.doi.org/10.1214/11-AAP759 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Concave Switching in Single and Multihop Networks
Switched queueing networks model wireless networks, input queued switches and
numerous other networked communications systems. For single-hop networks, we
consider a {()-switch policy} which combines the MaxWeight policies
with bandwidth sharing networks -- a further well studied model of Internet
congestion. We prove the maximum stability property for this class of
randomized policies. Thus these policies have the same first order behavior as
the MaxWeight policies. However, for multihop networks some of these
generalized polices address a number of critical weakness of the
MaxWeight/BackPressure policies.
For multihop networks with fixed routing, we consider the Proportional
Scheduler (or (1,log)-policy). In this setting, the BackPressure policy is
maximum stable, but must maintain a queue for every route-destination, which
typically grows rapidly with a network's size. However, this proportionally
fair policy only needs to maintain a queue for each outgoing link, which is
typically bounded in number. As is common with Internet routing, by maintaining
per-link queueing each node only needs to know the next hop for each packet and
not its entire route. Further, in contrast to BackPressure, the Proportional
Scheduler does not compare downstream queue lengths to determine weights, only
local link information is required. This leads to greater potential for
decomposed implementations of the policy. Through a reduction argument and an
entropy argument, we demonstrate that, whilst maintaining substantially less
queueing overhead, the Proportional Scheduler achieves maximum throughput
stability.Comment: 28 page
Store-Forward and its implications for Proportional Scheduling
The Proportional Scheduler was recently proposed as a scheduling algorithm
for multi-hop switch networks. For these networks, the BackPressure scheduler
is the classical benchmark. For networks with fixed routing, the Proportional
Scheduler is maximum stable, myopic and, furthermore, will alleviate certain
scaling issued found in BackPressure for large networks. Nonetheless, the
equilibrium and delay properties of the Proportional Scheduler has not been
fully characterized.
In this article, we postulate on the equilibrium behaviour of the
Proportional Scheduler though the analysis of an analogous rule called the
Store-Forward allocation. It has been shown that Store-Forward has
asymptotically allocates according to the Proportional Scheduler. Further, for
Store-Forward networks, numerous equilibrium quantities are explicitly
calculable. For FIFO networks under Store-Forward, we calculate the policies
stationary distribution and end-to-end route delay. We discuss network
topologies when the stationary distribution is product-form, a phenomenon which
we call \emph{product form resource pooling}. We extend this product form
notion to independent set scheduling on perfect graphs, where we show that
non-neighbouring queues are statistically independent. Finally, we analyse the
large deviations behaviour of the equilibrium distribution of Store-Forward
networks in order to construct Lyapunov functions for FIFO switch networks
Qualitative Properties of alpha-Weighted Scheduling Policies
We consider a switched network, a fairly general constrained queueing network
model that has been used successfully to model the detailed packet-level
dynamics in communication networks, such as input-queued switches and wireless
networks. The main operational issue in this model is that of deciding which
queues to serve, subject to certain constraints. In this paper, we study
qualitative performance properties of the well known -weighted
scheduling policies. The stability, in the sense of positive recurrence, of
these policies has been well understood. We establish exponential upper bounds
on the tail of the steady-state distribution of the backlog. Along the way, we
prove finiteness of the expected steady-state backlog when , a
property that was known only for . Finally, we analyze the
excursions of the maximum backlog over a finite time horizon for . As a consequence, for , we establish the full state space
collapse property.Comment: 13 page
On the Flow-level Dynamics of a Packet-switched Network
The packet is the fundamental unit of transportation in modern communication
networks such as the Internet. Physical layer scheduling decisions are made at
the level of packets, and packet-level models with exogenous arrival processes
have long been employed to study network performance, as well as design
scheduling policies that more efficiently utilize network resources. On the
other hand, a user of the network is more concerned with end-to-end bandwidth,
which is allocated through congestion control policies such as TCP.
Utility-based flow-level models have played an important role in understanding
congestion control protocols. In summary, these two classes of models have
provided separate insights for flow-level and packet-level dynamics of a
network
Throughput and Delay Scaling in Supportive Two-Tier Networks
Consider a wireless network that has two tiers with different priorities: a
primary tier vs. a secondary tier, which is an emerging network scenario with
the advancement of cognitive radio technologies. The primary tier consists of
randomly distributed legacy nodes of density , which have an absolute
priority to access the spectrum. The secondary tier consists of randomly
distributed cognitive nodes of density with , which
can only access the spectrum opportunistically to limit the interference to the
primary tier. Based on the assumption that the secondary tier is allowed to
route the packets for the primary tier, we investigate the throughput and delay
scaling laws of the two tiers in the following two scenarios: i) the primary
and secondary nodes are all static; ii) the primary nodes are static while the
secondary nodes are mobile. With the proposed protocols for the two tiers, we
show that the primary tier can achieve a per-node throughput scaling of
in the above two scenarios. In the associated
delay analysis for the first scenario, we show that the primary tier can
achieve a delay scaling of
with . In the second scenario, with two mobility
models considered for the secondary nodes: an i.i.d. mobility model and a
random walk model, we show that the primary tier can achieve delay scaling laws
of and , respectively, where is the random walk
step size. The throughput and delay scaling laws for the secondary tier are
also established, which are the same as those for a stand-alone network.Comment: 13 pages, double-column, 6 figures, accepted for publication in JSAC
201
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